Fast Matrix Rank Algorithms and Applications

We consider the problem of computing the rank of an m x n matrix A over a field. We present a randomized algorithm to find a set of r = rank(A) linearly independent columns in \~O(|A| + r^\omega) field operations, where |A| denotes the number of nonzero entries in A and \omega < 2.38 is the matrix multiplication exponent. Previously the best known algorithm to find a set of r linearly independent columns is by Gaussian elimination, with running time O(mnr^{\omega-2}). Our algorithm is faster when r < max(m,n), for instance when the matrix is rectangular. We also consider the problem of computing the rank of a matrix dynamically, supporting the operations of rank one updates and additions and deletions of rows and columns. We present an algorithm that updates the rank in \~O(mn) field operations. We show that these algorithms can be used to obtain faster algorithms for various problems in numerical linear algebra, combinatorial optimization and dynamic data structure.